C1 alternative exam – non-calculator (3 hours)
You can complete as many questions as you like.
Your best question from each section will count towards your total mark, so make
sure you attempt at least one question from each section.
Each question is worth 25 marks.
Section A: Surds and Indices
Question A1
Part 1
Sketch the graphs of 𝑦 = 𝑥 and 𝑦 = 𝑥 2 on the same axes. Show carefully where
they meet.
Describe two similarities and two differences between these graphs.
Part 2
Evaluate 43/2 and (−4)3/2 .
Sketch the graph of 𝑦 = 𝑥 3/2 on the same axes as the graphs of 𝑦 = 𝑥 and 𝑦 =
𝑥 2 , again being careful to show where the graphs meet.
Part 3
At which point do the graphs 𝑦 = 𝑥 and 𝑦 = 𝑥 2 have the same gradient?
At which point do the graphs 𝑦 = 𝑥 and 𝑦 = 𝑥 3/2 have the same gradient?
Which graph is the steepest at 𝑥 = 1/2? Explain how you know.
Part 4
Consider all graphs of the form 𝑦 = 𝑥 𝑎 .
How does the graphs differ for those where 𝑎 is an integer compared to those where
𝑎 is not an integer?
Question A2
For this question, 𝑃 = 2𝑚 and 𝑄 = 3𝑛 where 𝑚 and 𝑛 are positive integers.
Part 1
For this part only, let 𝑚 = 3 and 𝑛 = 1.
Evaluate: (a) P
(b) 2P
(c) Q/3
(d) PQ
Part 2
We can express expressions involving 𝑃 and 𝑄 in terms of 2, 3, 𝑚 and 𝑛.
For example, 2𝑃 = 2. 2𝑚 = 2𝑚+1.
Find expressions in terms of 2, 3, 𝑚 and 𝑛 for the following:
(a) 𝑄/3
(b) 𝑄 2
(c) √𝑃
(d) 2𝑃𝑄
(e) 𝑃𝑛
Part 3
A student wrote: “𝑃𝑄 = 2𝑚 . 3𝑛 , so 𝑃𝑄 = 6𝑚𝑛 .”
Explain why this is not correct, using examples to make your explanations clear.
Can you find an expression in terms of 𝑃, 𝑄, 𝑚 and 𝑛 that is equal to 6𝑚𝑛 ?
Question A3
A ‘prime surd’ is a surd √𝑝 where 𝑝 is a prime number. So √2 and √7 are prime
surds, but √10 is not.
Every surd can be written as a product of prime surds, for example:
√10 = √2 × √5
√12 = √2 × √2 × √3
and
Part 1
Use the product of prime surds above to write √12 in form 𝑎√𝑏 where 𝑎 and 𝑏 are
integers.
Write √18 as the product of prime surds, and hence write √18 in the form 𝑎√𝑏.
Hence or otherwise write √216 in form 𝑎√𝑏.
Part 2
Is 10 − 3√11 positive or negative? Explain how you know.
1
Show that 10−3√11 = 10 + 3√11
How could you have used this answer to show whether 10 − 3√11 was positive or
negative?
Part 3
Find another set of three numbers a, b and c so that
1
𝑎 − 𝑏 √𝑐
= 𝑎 + 𝑏 √𝑐
Part 4
Which is biggest, 10 − 3√11 or 𝑎 + 𝑏√𝑐 for the numbers you found in Part 3?
Explain how you know.
Section B - quadratics
Question B1
Part 1
The equation 𝑥 2 – 22𝑥 + 57 = 0 has two prime roots – can you find them?
Given that both roots of the quadratic 𝑥 2 – 22𝑥 + 𝑘 = 0 are prime roots, how many
possible values of 𝑘 are there?
Given that both roots of the quadratic 𝑥 2 – 99𝑥 + 𝑘 = 0 are prime, how many
possible values of k are there?
Part 2
Daisy said: “If any quadratic expression of the form 𝑛2 + 𝑎. 𝑛 + 𝑏 is factorisable,
then it can only be prime for exactly one value of n.”
Factorise the quadratic 𝑛2 − 3𝑛 + 2.
Show that there is only one value of 𝑛 that makes this expression prime.
Do you think the student’s statement is always true? Give reasons for your answers.
Part 3
Daisy then said: “The expression 𝑛2 + 𝑛 + 41 comes up with lots of primes. For
example, substituting 𝑛 = 1, 2, 3, … into this one gives: 43, 47, 53, … which are all
prime. I think this quadratic might generate primes for ever!”
Do you think she is correct? Explain your answer.
Question B2
Let 𝑎 and 𝑏 be a pair of different numbers from the set {−2, −1, 1, 2}.
We are going to use the two numbers 𝑎 and 𝑏 to make quadratic equations of the
form 𝑥 2 + 𝑎𝑥 + 𝑏 = 0.
For example, we can choose 𝑎 = −2 and 𝑏 = −1 to make the equation:
𝑥 2 − 2𝑥 − 1 = 0
Part 1
How many different equations can we make?
Part 2
Which of these equations have no real roots?
Which equations have equal roots?
Which equations have two different real roots?
Part 3
Which equation has two roots with a sum of 1?
Part 4
Which equations have roots with a difference of 2√2?
Part 5
Which equation of the form 𝑥 2 + 𝑎𝑥 + 𝑏 = 0 has roots 𝑎 and 𝑏?
Question B3
Part 1
Write the quadratic 𝑥 2 – 6𝑥 + 9 in the form (𝑥 + 𝑝)2 + 𝑞, where 𝑝 and 𝑞 are
integers to be found.
Sketch the graph of 𝑦 = 𝑥 2 – 6𝑥 + 9, showing where the graph crosses the
coordinate axes, and the coordinates of the vertex.
How does completing the square help you draw the graph 𝑦 = 𝑥 2 – 6𝑥 + 9?
Now sketch the graphs of 𝑦 = 𝑥 2 – 6𝑥 + 8 and 𝑦 = 𝑥 2 – 6𝑥 + 10, being sure to
show where the graphs cross the axes and the position of the vertex.
Part 2
A student solved the equation 𝑥 2 − 2𝑥 − 1 = 0 incorrectly as follows:
𝑥 2 − 2𝑥 − 1 = 0
(𝑥 − 1)2 = 0
𝑥=1
Show where they made a mistake, and give a correct solution.
Part 3
Find a quadratic that has solutions 1 + √3 and 1 − √3.
Find a quadratic that has solutions 2 + √3 and 2 − √3.
Find the general form of the quadratic equation (i.e. in terms of m and n) that has
solutions 𝑚 + √𝑛 and 𝑚 − √𝑛.
Hence, or otherwise, solve the quadratic 𝑥 2 – 20𝑥 + 89 = 0.
Question B4
In this question, a quadratic is said to be factorisable if it can written in the form
(𝑥 + 𝑎)(𝑥 + 𝑏) where 𝑎 and 𝑏 are integers.
Part 1
Erin said: “A quadratic 𝑥 2 + 𝑏𝑥 + 𝑐 is factorisable if and only if
𝑏 2 – 4𝑐 is a square number.”
Are they partially correct or fully correct? Explain how you know.
Part 2
Fariha said: “If we can factorise 𝑥 2 + 𝑏𝑥 + 𝑐, then we can always also factorise
𝑥 2 – 𝑏𝑥 + 𝑐. For example, we can factorise both 𝑥 2 – 4𝑥 + 3 and 𝑥 2 + 4𝑥 + 3.”
Is this always true? Explain how you know.
Can you use Erin’s statement to help?
Part 3
Gemma said: “If we can factorise 𝑥 2 + 𝑏𝑥 + 𝑐, then we can never factorise 𝑥 2 +
𝑏𝑥 − 𝑐.”
Is this always true? Explain how you know.
Part 4
Harry said: “If the b and c in 𝑥 2 + 𝑏𝑥 + 𝑐 are both odd prime numbers then the
quadratic is not factorisable.”
Is this always true? Explain how you know.
Section C: Functions
Question C1
Part 1
Consider the function 𝑓(𝑥) = 𝑥(𝑥 – 3)(𝑥 + 2).
(a) Sketch the function 𝑓(𝑥 + 1), showing where the graph crosses the coordinate
axes.
(b) Sketch the function 𝑓(−𝑥) on the same graph as 𝑓(𝑥 + 1), again showing where
the graph crosses the coordinate axes.
(c) Where do 𝑓(𝑥 + 1) and 𝑓(−𝑥) intersect? Show your working.
Part 2
Solve these inequalities:
(a) 𝑓(𝑥) > 0
(b) 𝑓(𝑥 + 1) > 0
(c) 𝑓(−𝑥) > 0
Part 3
Hadar was investigating graphs of cubics of the form 𝑦 = 𝑥 3 + 𝑝𝑥 + 𝑞.
He noticed that if 𝑝 > 0 then the graph only crosses the x-axis once.
Do you think he is correct? Can you explain why?
How many times does a cubic cross the x-axis if 𝑝 < 0?
Question C2
Part 1
Let 𝑓(𝑥) = 2𝑥 2 − 𝑥 – 10.
Solve the equation 𝑓(𝑥) = 0.
Hence, or otherwise, find the integer values of x for which:
(a) 𝑓(𝑥) < 0
(b) 2. 𝑓(𝑥) < 0
(c) 𝑓(2𝑥) < 0
(d) 𝑓(𝑥 − 2) < 0
Part 2
Let 𝑔(𝑥) = 𝑥 2 – 10.
Find the integer values for which:
(a) 𝑔(𝑥) < 0
𝑥
(b) 𝑔 ( ) < 0
√2
(c) 𝑔(𝑥) + 1 < 0
Part 3
For which values of 𝑥, not necessarily integers, is 𝑔(𝑥) ≥ 𝑓(𝑥)?
Sketch the two graphs on the same axes, being careful to show how they cross, and
where they meet the coordinate axes.
Question C3
Andrew was exploring the effects of function transformations.
Part 1
He started with the function 𝑓(𝑥) = 2𝑥 + 1 and then
worked out the algebraic form of the function 𝑓(𝑥 + 1) as
follows:
𝑓(𝑥 + 1) = 2(𝑥 + 1) + 1 = 2𝑥 + 3
He then said: “This is 2 higher than 2𝑥 + 1, so the
transformation 𝑓(𝑥 + 1) shifts the graph up.”
Which parts of Andrew’s statement are fully correct are
which parts are only partially correct?
Give an example to show that the transformation 𝑓(𝑥 + 1) does not translate every
function up by 2.
Write a description of a more useful way to think about this transformation. Make
sure you include reasons why it is better to think of this transformation in this way.
Part 2
Andrew was then thinking about the transformation 𝑓(2𝑥). He again performed the
algebraic transformation, like this:
𝑓(2𝑥) = 2(2𝑥) + 1 = 4𝑥 + 1
He then concluded that the transformation 𝑓(2𝑥) “doubles the gradient but keeps
everything else the same”.
Which parts of Andrew’s statement are fully correct are which parts are only
partially correct?
Can you describe a more useful way to think about this transformation? Be sure to
give examples to clarify your explanation.
Section D: coordinate geometry
Question D1
Part 1
What is the gradient of the line containing the points (2,1) and (1,2)?
What is the equation of this line?
Part 2
What is the gradient of the line containing the points (𝑎, 1) and (1, 𝑎)?
What is the equation of this line, giving the y-intercept in terms of 𝑎?
Part 3
The points (𝑏, 3) and (1, 𝑏) lie on the line with gradient 1.
What is the value of 𝑏?
What is the equation of the line?
Part 4
The points (𝑚, 3) and (1, 𝑚) lie on the line with gradient 𝑚.
What is the value of 𝑚?
What is the equation of this line?
Find all the points with integer coordinates that this line passes through.
Question D2
The graph of a line with equation 𝑦 = 𝑚𝑥 + 𝑐 is shown.
Part 1
Which of the following is true for this line?
(A) 𝑚𝑐 < −1
(B) −1 < 𝑚𝑐 < 0
(C) 𝑚𝑐 = 0
(D) 0 < 𝑚𝑐 < 1
(E) 𝑚𝑐 > 1
Part 2
For a different line with 𝑚𝑐 > 1, the line with equation 𝑦 = 𝑚𝑥 + 𝑐 cannot contain
which one of these points:
(A) (0,2015)
(B) (0, −2015)
(D) (20, −15)
(E) (2015,0)
(C) (20,15)
Part 3
Draw a line that would match each ‘false’ expression in Part 1, being careful to make
the value of the x- and y-intercepts clear for each line.
Question D3
Part 1
The triangle ABC has vertices 𝐴 = (0,0), 𝐵 = (3,0) and 𝐶 = (1,4). What is the area
of the triangle ABC?
Part 2
The triangle DEF has vertices 𝐷 = (0,0) and 𝐸 = (0,4). The third vertex F lies on the
line 𝑥 = 3. What is the area of the triangle DEF?
Part 3
The triangle GHJ has vertices 𝐺 = (3,0), 𝐻 = (0,3). The third vertex J lies on the line
𝑥 + 𝑦 = 7. What is the area of the triangle GHJ?
Part 4
A point A is chosen on the line 𝑦 = 12/𝑥. A vertical is then dropped to the x-axis,
and a triangle is formed with the origin like this:
Show that the area of the triangle is not dependent on the choice of P, and find the
area of the triangle.
Find the coordinates of the two points on the line which give a triangle with
perimeter double the area.
Section E: sequences and series
Question E1
Part 1
The first term of an arithmetic sequence is 4, and the sum of the first three terms is
27. Find each term.
Now, the first term of a different arithmetic sequence is 5, and the sum of the first
three terms is 27. Find each term.
If the sum of the first three terms of any arithmetic sequence is 27, can you work out
any of the terms?
Part 2
The sum of the first 9 terms of an arithmetic sequence is 99.
Can you work out any of the terms?
If the last term is 23, what is the first term?
Part 3
The first three terms of an arithmetic sequence sum to 30, and the next three terms
sum to 57.
What is the nth term of the sequence?
Question E2
Part 1
Here are the 3rd and 4th terms of an arithmetic sequence:
…,
3
√2
, 2√2 , …
What are the 1st, 2nd and 5th terms?
What is the 100th term?
Part 2
Find the sum of the first five terms, giving your answer in the form 𝑎√2.
Part 3
Prove that the first term of this sequence is irrational.
How many rational numbers are there in this sequence? How do you know?
Part 4
5
12
Which is the best approximation to the first term, 7 or 17 ?
Explain how you know.
Question E3
This question is based on Pascal’s triangle.
It might help you answer some of the
questions.
Part 1
What is the sum of the first 10 counting numbers (1, 2, 3, … , 10)?
What is the sum of the first 10 even counting numbers (2, 4, … , 20)?
What is the sum of the first 10 odd counting numbers?
What is the difference between the sum of the first 2015 even counting numbers,
and the sum of the first 2015 odd counting numbers?
Part 2
Where are the triangle numbers in Pascal’s triangle?
Why do they appear?
What is the sum of the first 10 triangle numbers?
Take any pair of consecutive triangle numbers and add them. What do you notice?
Can you explain why this is true?
Part 3
Work out the total of each row in Pascal’s triangle.
Write down a recurrence relationship for the totals in each row.
Can you explain why this relationship is true?
Section F: Calculus
Question F1
Part 1
Let 𝑦 = 2√𝑥, where 𝑥 > 0.
𝑑𝑦
Evaluate 𝑑𝑥 at 𝑥 = 1, 𝑥 = 2 and 𝑥 = 3.
What can you say about the gradient of 𝑦 = 2√𝑥 as 𝑥 increases?
Sketch the graph of 𝑦 = 2√𝑥.
Do you think this curve has a horizontal asymptote? Explain your answer.
Part 2
Find the equations of the normal to the curve 𝑦 = 2√𝑥 at 𝑥 = 1, 𝑥 = 2 and 𝑥 = 3 in
the form 𝑦 + 𝑎𝑥 = 𝑏.
Express the y-intercept and x-intercept of the normal to the curve 𝑦 = 2√𝑥 at 𝑥 = 𝑝
in terms of 𝑝.
Find the area of the shape enclosed by the x-axis, the y-axis and the normal to the
curve 𝑦 = 2√𝑥 at 𝑥 = 9.
Question F2
Let 𝑓(𝑥) =
1+𝑥
𝑥
, where 𝑥 > 0.
Part 1
Substitute at least three values of 𝑥 into 𝑓(𝑥) and sketch the curve.
Make sure you consider carefully what happens for small and large values of 𝑥.
What is the equation of the horizontal asymptote for this curve?
Part 2
Complete this table of differences by filling in the un-shaded squares.
The first few are done for you:
𝑥
1
2
3
𝑓(𝑥)
2
3/2
4/3
Differences:
4
5
-1/2
Example: The difference of −1/2 was calculated from 𝑓(2) – 𝑓(1) = 3/2 – 2.
What happens to the differences, as 𝑥 gets larger?
Part 3
Find 𝑓’(𝑥).
Work out 𝑓’(𝑥) for the values of 𝑥 shown in the table.
Compare them with the differences in the table. What do you notice?
Explain how you know that the gradient of this curve will never be greater than or
equal to zero.
Question F3
A student was investigating tangents to 𝑦 = 𝑥 2 using Geogebra. They plotted the
tangent at 𝑥 = 2 as shown below:
The student then made the following conjectures:
“It looks like the tangent is ‘symmetrical’ in some way. For example, the tangent at
(2,4) crosses the y-axis at -4, so these y-values are kind of ‘opposites’. This might
always be true, but it might just be a coincidence.
Also, the tangent looks like it cuts through the x-axis at 1, which is halfway between 0
and 2. I think this might always be true as well, although I’m not sure why.”
Use calculus to explore the student’s conjectures for tangents at other points on
𝑦 = 𝑥2.
Question F4
Part 1
Give an example of an expression that has derivative 𝑥 2 . Now give a different
example.
Which function 𝑓(𝑥) with derivative 𝑓 ′ (𝑥) = 𝑥 2 passes through the point (−2, 0)?
Part 2
What is the gradient of 𝑓(𝑥) at the point (−2, 0)?
It which other point on 𝑓(𝑥) does the gradient have the same value?
For which values of x will the gradient be negative?
Part 3
Sketch 𝑦 = 𝑓(𝑥).
Describe the symmetry of the graph.
Part 4
Answer Parts 1 to 3 of this question again, but with derivative 𝑓 ′ (𝑥) = 𝑥 3 .